For-profit companies should be brought in to help improve hundreds of underperforming state schools, according to the former headteacher of an inner-city comprehensive.

Trevor Averre-Beeson, ex-head at Islington Green comprehensive in north London, now works as the education director of one of the UK's biggest for-profit education firms, Lilac Sky Schools.

He said many of the 1,310 primary schools and 107 secondaries deemed underperforming could be "transformed" by companies such as his own.

The debate over whether for-profit firms should have a greater role in running schools is highly political and is expected to intensify this week when Policy Exchange, a right-of-centre thinktank, publishes a report calling on ministers to go further in allowing for-profits into state schools.

Last year Michael Gove, the education secretary, said he did not have a "particular objection to involving any organisation that is going to improve our education. [But] we don't need to have profit-making organisations involved at the moment."

However, many Conservatives are said to be pressing for the next party manifesto to include ideas on how to allow for-profit firms more involvement in schools. Some argue that the companies should be given the right to set up free schools and claim this would increase competition and therefore improve school standards.

Nick Clegg has said the Lib Dems are opposed to running schools for profit.

A growing number of English schools are working with for-profit companies. Last month Gove gave the green light for Breckland middle school in Suffolk to be renamed IES Breckland and run under a £21m, 10-year contract by the Swedish for-profit firm Internationella Engelska Skolan.

The government's preferred model for underperforming schools is to turn them into academies. The trusts of academy schools must be established on a not-for-profit basis and the trusts can let contracts for the running of parts or all of a school's services. As long as a full EU procurement is carried out, this contract can be let for-profit. The same applies to non-academy schools.

Averre-Beeson said for-profit companies could just as easily work with failing schools that were not academies as schools that were.

"Becoming an academy is a solution for the majority of under-performing schools, but when there is a strong desire to maintain the character of a school … the solution could be for a company to run the school on a contracted-out basis for a number of years," he said. His company has contracts with 21 UK schools and is negotiating several more.

Averre-Beeson said for-profit companies were more focused on improving a school than a new headteacher would be because they had a contract with specific targets to reach. He said they were more accountable to local communities and governing bodies as a result.

"There are some schools that have been closed down and reopened. That is one solution and it works in the right circumstances, but where the local community doesn't want a school to close, it could enter into a contract with a private organisation. Schools don't have to go down an academy route," he said.

Melissa Benn, an education campaigner and writer, said parents and the public should be given the right to vote on whether for-profits should have more of a say in running schools. "This is such a big step-change," she said. "If we lose the idea of education as a public good, we lose something very valuable."

A forthcoming report by the Institute for Public Policy Research, a centre-left thinktank, has found that across the world, for-profit education companies have at best a mixed record.

The IPPR's study – Not for Profit: The role of the private sector in England's schools – says there have only been large-scale experiments of for-profits running schools in Sweden, Chile and some US states. In Michigan and Florida, studies show that for-profits did not run schools any better than not-for-profit providers did. In Sweden and Chile, not-for-profit firms ran schools better than for-profit ones.

"There is no evidence that you need for-profit companies to introduce innovation into education," said the report's author, Rick Muir.

David Bell, who was permanent secretary at the Department for Education until the end of 2011, told the Guardian this month that he saw "no principled objection" to profit-making companies taking over state schools and expected they probably would do eventually.